A temperature measurement system, assumed to exhibit a first-order response with a 90%
risetime of 8s and static sensitivity of 5mV/°C, is used to measure the temperature of various
industrial processes. Answer the following questions regarding this system:

If the system is used to measure the temperature of a flowing gas the varies sinusoidally
from 20°C to 100° with a period of 30s:
a. Find the average value of the temperature waveform



The temperature waveform described in the problem varies sinusoidally between 20°C and 100°C with a period of 30 seconds. To find the average value of this sinusoidal temperature waveform, we need to calculate the mean (average) of the waveform over a full cycle.

For a sinusoidal waveform, the average value over one complete cycle is equal to the mean of the maximum and minimum values. This is a known property of sinusoidal waveforms.

### Step-by-Step Calculation

The temperature waveform oscillates between:
- **Maximum temperature**: \( T_{max} = 100°C \)
- **Minimum temperature**: \( T_{min} = 20°C \)

The average value of the sinusoidal waveform over a full period is given by:

\[
T_{avg} = \frac{T_{max} + T_{min}}{2}
\]

Let's calculate it.

\[
T_{avg} = \frac{100°C + 20°C}{2} = \frac{120°C}{2} = 60°C
\]

### Conclusion

The average value of the sinusoidal temperature waveform is **60°C**.

Would you like more details or have any further questions?

### Related Questions:
1. How would the average value change if the temperature oscillated between different limits?
2. What is the significance of the average temperature in industrial processes?
3. How does the frequency of oscillation affect the system's response in a first-order system?
4. What is the output voltage corresponding to the average temperature in this system?
5. How would the system behave if the period of the temperature oscillation was shorter than the system's time constant?

### Tip:
For sinusoidal signals, the average value over one period is simply the midpoint between the maximum and minimum values.

Learn how to find the average value of a sinusoidal temperature waveform oscillating between 20°C and 100°C with a period of 30 seconds. Understand the mathematical approach to averaging sinusoidal signals in temperature measurement systems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation